┌ Warning: Found `resolution` in the theme when creating a `Scene`. The `resolution` keyword for `Scene`s and `Figure`s has been deprecated. Use `Figure(; size = ...` or `Scene(; size = ...)` instead, which better reflects that this is a unitless size and not a pixel resolution. The key could also come from `set_theme!` calls or related theming functions.
└ @ Makie ~/.julia/packages/Makie/gG38B/src/scenes.jl:227
Example continued: how fast does the solution approach the origin?
Let’s use the 1-norm \|\bm x\|_1 = |x_1| + |x_2| to measure how far the trajectory is from the origin.
How fast does the trajectory converge to the origin? That, is
\frac{\mathrm d}{\mathrm dt}\|\bm x\|_1 = ?
Consider each quadrant separately. Let’s start in the first (upper right) quadrant, that is, x_1>0 and x_2>0, and therefore |x_1| = x_1, \;|x_2| = x_2, and therefore
Instead solving the above nasty equations, introduce new variables u_1 and u_2 as the outcomes of the \operatorname{sign} functions:
\begin{aligned}
{\color{blue} x_{1,k+1}} &= x_{1,k} + h (-{\color{blue}u_{1}} + 2 {\color{blue}u_{2}})\\
{\color{blue} x_{2,k+1}} &= x_{1,k} + h (-2{\color{blue}u_{1}} - {\color{blue}u_{2}})
\end{aligned}
But now we have to enforce the relation between \bm u and \bm x_{k+1}.
Recall the standard definition of the \operatorname{sign} function:
\operatorname{sign}(x) = \begin{cases}
1 & x>0\\
0 & x=0\\
-1 & x<0
\end{cases}
Change the definition to a set-valued function
\begin{cases}
\operatorname{sign}(x) = 1 & x>0\\
\operatorname{sign}(x) \in [-1,1] & x=0\\
\operatorname{sign}(x) = -1 & x<0
\end{cases}
Accordingly, set the relation between \bm u and \bm x
\begin{cases}
u_1 = 1 & x_1>0\\
u_1 \in [-1,1] & x_1=0\\
u_1 = -1 & x_1<0
\end{cases}
and
\begin{cases}
u_2 = 1 & x_2>0\\
u_2 \in [-1,1] & x_2=0\\
u_2 = -1 & x_2<0
\end{cases}
How does the set of states from which the next state is zero look like?
\begin{aligned}
-\begin{bmatrix}
-1 & 2 \\
-2 & -1
\end{bmatrix}^{-1}
\begin{bmatrix}
x_{1,k}\\
x_{2,k}
\end{bmatrix}
&= h
\begin{bmatrix}
{\color{blue}u_{1}}\\
{\color{blue}u_{2}}
\end{bmatrix}\\
-1 \leq {\color{blue} u_1} \leq 1, \quad -1 &\leq {\color{blue} u_2} \leq 1
\end{aligned}
